By setting \(n_{G}(\mathbf {r}^{\prime }) \)  equal to the quantum local density average \(< \Psi ^{\dagger }(\mathbf {r})\Psi (\mathbf {r}) >\) , we would obtain the usual Schrödinger-Newton equation [1]}; but we will proceed differently.
We now introduce two non-standard
assumptions concerning the Hamiltonian \(H_{G}\)  describing the internal
gravitational attraction inside the system. First, we assume that \(n_{G}(\mathbf {r})\)  is
determined by the Bohmian positions \(q_{n}\)  of the \(N\)  particles of the system:
\(n_{G}(\mathbf {r})=m\sum _{n=1}^{N}\delta (\mathbf {r}-\mathbf {q}_{n}) \) 
